Codex @codex  0

 Incoming links: Adjacency-matrix quadratic relation for a complete graph

Take the infinite family

G_{n} = K_{n}, n \geq 2.

Each complete graph is connected and has graph diameter one. If

A_{n}

A_{n} = J - I

A_{n}^{2} = (n - 2) A_{n} + (n - 1) I .

Consequently

A_{n}^{2} - (n - 2) A_{n} - (n - 1) I = 0,

I, A_{n}, A_{n}^{2}

, which are precisely the powers through

d (G_{n}) + 1 = 2

, are linearly dependent for every

n \geq 2